# Rigged Hilbert space

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A Hilbert space containing a linear, everywhere-dense subset , on which the structure of a topological vector space is defined, such that the imbedding is continuous. This imbedding generates a continuous imbedding of the dual space and a chain of continuous imbeddings (using the standard identification ). The most interesting case is that in which is a nuclear space. The following strengthening of the spectral theorem for self-adjoint operators acting on is true: Any self-adjoint operator mapping continuously (in the topology of ) onto itself possesses a complete system of generalized eigenfunctions ( is a set of indices), i.e. elements such that for any , where the set of values of the function , , is contained in the spectrum of (cf. Spectrum of an operator) and has full measure with respect to the spectral measure , , , of any element . The completeness of the system means that for any , , for at least one . Moreover, for any element , its expansion with respect to the system of generalized eigenfunctions exists and generalizes the known expansion with respect to the basis of eigenvectors for an operator with a discrete spectrum.

Example: The expansion into a Fourier integral  is a system of generalized eigenfunctions of the differentiation operator, acting on , arising under the natural rigging of this space by the Schwartz space (cf. Generalized functions, space of). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.

How to Cite This Entry:
Rigged Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rigged_Hilbert_space&oldid=13083
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article